dc.contributor.author Bai, Ruobing dc.contributor.author Wu, Yifei dc.contributor.author Xue, Jun dc.date.accessioned 2021-03-05T15:20:08Z dc.date.available 2021-03-05T15:20:08Z dc.date.created 2020-09-29T14:19:10Z dc.date.issued 2020 dc.identifier.citation Journal of Differential Equations. 2020, 269 (9), 6422-6447. en_US dc.identifier.issn 0022-0396 dc.identifier.uri https://hdl.handle.net/11250/2731961 dc.description.abstract In this work, we consider the following generalized derivative nonlinear Schr\"odinger equation \begin{align*} i\partial_t u+\partial_{xx} u +i |u|^{2\sigma}\partial_x u=0, \quad (t,x)\in \R\times \R. \end{align*} We prove that when $\sigma\ge 2$, the solution is global and scattering when the initial data is small in $H^s(\R)$, $\frac 12\leq s\leq1$. Moreover, we show that when $0<\sigma<2$, there exist a class of solitary wave solutions $\{\phi_c\}$ satisfying $$\|\phi_c\|_{H^1(\R)}\to 0,$$ when $c$ tends to some endpoint, which is against the small data scattering statement. Therefore, in this model, the exponent $\sigma\ge2$ is optimal for small data scattering. We remark that this exponent is larger than the short range exponent and the Strauss exponent. en_US dc.language.iso eng en_US dc.publisher Elsevier en_US dc.rights Attribution-NonCommercial-NoDerivatives 4.0 Internasjonal * dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/4.0/deed.no * dc.title Optimal small data scattering for the generalized derivative nonlinear Schrödinger equations en_US dc.type Peer reviewed en_US dc.type Journal article en_US dc.description.version acceptedVersion en_US dc.source.pagenumber 6422-6447 en_US dc.source.volume 269 en_US dc.source.journal Journal of Differential Equations en_US dc.source.issue 9 en_US dc.identifier.doi https://doi.org/10.1016/j.jde.2020.05.001 dc.identifier.cristin 1834976 dc.relation.project Norges forskningsråd: 250070 en_US dc.description.localcode © 2020. This is the authors’ accepted and refereed manuscript to the article. Locked until 7/6-2022 due to copyright restrictions. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ en_US cristin.ispublished true cristin.fulltext postprint cristin.qualitycode 2
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